Introduction to Systems

When One Equation Is Not Enough

Until now, we have studied differential equations involving a single unknown function. A cooling cup of coffee, a decaying radioactive sample, an oscillating spring: each phenomenon involved one quantity changing over time.

But the world is full of interacting quantities. Predators eat prey. Competing species share resources. In an epidemic, susceptible people become infected, and infected people recover. The rate of change of one quantity depends not just on itself, but on others.

To model such interactions, we need systems of differential equations: multiple equations describing how multiple quantities evolve together, each influencing the others.

The Classic Example: Predator and Prey

Imagine an ecosystem with rabbits and foxes. Rabbits reproduce when food is plentiful, and foxes eat rabbits. Left alone with abundant grass, the rabbit population would grow exponentially. Left without food, the fox population would die off exponentially. But they do not exist in isolation: they interact.

More rabbits means more food for foxes, so foxes thrive. But more foxes means more predation, so rabbits decline. Fewer rabbits means less food for foxes, so foxes decline. Fewer foxes means less predation, so rabbits recover. And the cycle begins again.

This leads to the Lotka-Volterra equations, one of the most famous systems in mathematical biology:

Here $x$ represents the prey population (rabbits) and $y$ the predator population (foxes). The constants $a, b, c, d$ are all positive. Let us read what each term means:

• $ax$: prey grow naturally at rate $a$ when predators are absent
• $-bxy$: predation reduces prey; the rate depends on encounters between predators and prey
• $-cy$: predators die naturally at rate $c$ when prey are absent
• $dxy$: predators grow by consuming prey; growth depends on encounters

Notice how the populations oscillate out of phase. When prey is abundant, predators flourish shortly after. But as predators increase, prey declines. The predator boom is followed by a predator bust as their food source dwindles. Then prey recovers, and the cycle repeats.

In the phase plane view, we see something remarkable: the trajectory forms a closed loop. Each point on this loop represents a particular combination of prey and predator populations at some moment in time. The system cycles endlessly through these states, and time itself becomes invisible. What matters is the relationship between the two populations.

Other Systems in Nature

The predator-prey model is just one example. Systems of differential equations appear throughout science:

Competing species share a common resource. If two species compete for the same food, each suppresses the other's growth:

The parameters $\alpha$ and $\beta$ measure how strongly each species affects the other. Depending on these values, one species may drive the other to extinction, or they may coexist at some equilibrium.

Epidemic models track the flow of individuals between compartments. The SIR model divides a population into Susceptible ($S$), Infected ($I$), and Recovered ($R$):

Susceptible people become infected through contact with infected people (the $\beta SI$ term). Infected people recover at rate $\gamma$. This simple model captures the essential dynamics of how diseases spread through populations and eventually burn out.

Coupled oscillators appear in physics and engineering. Two pendulums connected by a spring, two circuits connected by a capacitor, two neurons connected by a synapse: all follow systems of differential equations describing how energy flows between the coupled components.

From Higher Order to Systems

There is a beautiful connection between higher-order equations and systems. Every $n$th-order differential equation can be converted into a system of $n$ first-order equations.

Consider the damped harmonic oscillator:

This single second-order equation involves $y$, $y'$, and $y''$. To convert it to a system, we introduce a new variable for each derivative up to (but not including) the highest one. Let $v = y'$. Then $v' = y''$, and our equation becomes:

Rearranging and writing as a system:

The single second-order equation has become two coupled first-order equations. The position $y$ changes according to the velocity $v$. The velocity $v$ changes according to the restoring force $-y$ and the damping $-0.3v$.

This conversion is not just a mathematical trick. It reveals a deeper truth: the state of a second-order system requires two pieces of information (position and velocity), just as a system of two first-order equations has a two-dimensional state space.

Vector Notation

Writing systems equation by equation becomes tedious. We can express them more compactly using vectors and matrices.

Define the state vector $\mathbf{x}$ to hold all unknown functions:

For a linear system, the right-hand side depends linearly on the state:

where $A$ is a constant matrix. This is the matrix analog of the scalar equation $y' = ky$.

For our damped oscillator, the matrix form is:

How matrix-vector multiplication works: If you have not seen this before, the product $A\mathbf{x}$ is computed row by row. For $\mathbf{x} = \begin{bmatrix} y \\ v \end{bmatrix}$:

So the matrix equation $\mathbf{x}' = A\mathbf{x}$ unpacks to exactly our two scalar equations: $y' = v$ and $v' = -y - 0.3v$.

This notation is not just convenient. The eigenvalues and eigenvectors of $A$ will tell us everything about the behavior of solutions, as we will see in later chapters. (Eigenvalues are special numbers that characterize how the matrix stretches or rotates vectors; eigenvectors are the special directions where the matrix acts simply by scaling.)

Nonlinear systems like Lotka-Volterra have the form $\mathbf{x}' = \mathbf{f}(\mathbf{x})$, where the function $\mathbf{f}$ may involve products, powers, or other nonlinear combinations of the state variables.

The Phase Plane

For a two-dimensional system, we can visualize all solutions at once using the phase plane. Each point $(x, y)$ represents a possible state of the system. At each point, the differential equation tells us the direction of motion: if the state is $(x, y)$, it moves with velocity $(x', y')$.

Drawing a small arrow at each point showing this velocity gives us a vector field. The arrows show how any trajectory would flow through that region of the phase plane.

The vector field is like a current map for solutions. Drop a leaf anywhere, and watch it flow along a trajectory. The pattern of the field reveals the qualitative behavior of all solutions at once.

Notice how different matrices produce qualitatively different flows:

• Stable spirals: trajectories wind inward toward the equilibrium
• Saddle points: trajectories approach along one direction, flee along the perpendicular
• Centers: trajectories orbit in closed loops, neither approaching nor fleeing

Understanding these qualitative behaviors, and connecting them to the properties of the matrix $A$, is a central goal of this part of the course.

Trajectories and Initial Conditions

A trajectory is the path traced by a solution in the phase plane. Given an initial condition $(x_0, y_0)$ at time $t = 0$, the solution $(x(t), y(t))$ traces out a curve as $t$ increases.

Different initial conditions give different trajectories. But for a given system, all trajectories share a common structure. They all follow the vector field, flowing through the phase plane like streams following the terrain.

Click to place initial conditions and watch trajectories unfold. Notice that trajectories never cross: if two solutions passed through the same point, the vector field would give them the same direction, so they would be the same solution. This uniqueness property means the phase plane is partitioned into non-intersecting curves.

What Lies Ahead

This chapter has introduced the key ideas:

• Systems model interacting quantities
• Higher-order equations convert to first-order systems
• Linear systems have the form $\mathbf{x}' = A\mathbf{x}$
• The phase plane visualizes all solutions via the vector field
• Trajectories are the paths solutions trace through state space

In the chapters that follow, we will develop tools to analyze systems systematically:

Phase Portraits will reveal the complete qualitative picture of a system's behavior, showing all trajectories and their asymptotic fates.

The Eigenvalue Method will show how the eigenvalues and eigenvectors of $A$ give explicit formulas for solutions of linear systems.

Stability and Classification will categorize equilibrium points as nodes, spirals, saddles, or centers based on eigenvalue properties.

The bridge from single equations to systems opens up a much richer world. We can finally model phenomena where multiple quantities evolve together, influencing each other in intricate ways.

Key Takeaways

• Systems of differential equations model interacting quantities where each rate of change depends on multiple variables
• The Lotka-Volterra predator-prey equations produce oscillating populations that cycle endlessly in the phase plane
• Any $n$th-order ODE can be converted to a system of $n$ first-order equations by introducing variables for each derivative
• Linear systems have the compact form $\mathbf{x}' = A\mathbf{x}$, where solutions are determined by the matrix $A$
• The phase plane shows the vector field: arrows indicating the direction of motion at each point
• Trajectories are curves traced by solutions; different initial conditions give different trajectories that never cross
• The geometry of the vector field reveals qualitative behavior: spirals, saddles, centers, and more